Influence of aperture field heterogeneity and anisotropy on dispersion regimes and dispersivity in single fractures
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1 McMaster University From the SelectedWorks of Sarah E Dickson 2009 Influence of aperture field heterogeneity and anisotropy on dispersion regimes and dispersivity in single fractures Qinghuai Zheng, McMaster University Sarah E Dickson, McMaster University Yiping Guo, McMaster University Available at:
2 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2007jb005161, 2009 Influence of aperture field heterogeneity and anisotropy on dispersion regimes and dispersivity in single fractures Q. Zheng, 1 S. Dickson, 1 and Y. Guo 1 Received 4 May 2007; revised 11 July 2008; accepted 5 December 2008; published 12 March [1] A3 3 factorial experimental design was implemented to numerically investigate the interactive effect of the mean (m b ), standard deviation (s b ), and anisotropic ratio (AR) (l b x/l b y) of single-fracture apertures on dispersion regimes (specifically Taylor dispersion and geometric dispersion) and dispersivity. The Reynolds equation was solved to obtain the flow fields in each computer-generated fracture, and the random walk particle tracking method was used to simulate solute transport. The simulation results show that (1) for a fixed hydraulic gradient, the dominant dispersion regime is controlled by m b, and to a lesser degree, s b, and geometric dispersion becomes more dominant as the coefficient of variation (COV) (s b /m b ) increases; (2) for a fixed mean aperture, the dispersivity and the spread in dispersivity for varying ARs increases with the COV; and (3) the AR has a significant effect on dispersivity only when the COV is large (^0.2). Citation: Zheng, Q., S. Dickson, and Y. Guo (2009), Influence of aperture field heterogeneity and anisotropy on dispersion regimes and dispersivity in single fractures, J. Geophys. Res., 114,, doi: /2007jb Introduction [2] The geometry of aperture fields and the morphology of fracture walls are among the primary factors influencing flow and transport in single fractures; however, the relative importance of these properties remains to be evaluated [Gentier et al., 1997; Meheust and Schmittbuhl, 2001]. Aperture field measurement using destructive (e.g., surface topography, casting, and injection) and nondestructive (e.g., magnetic resonance, X-ray, and light transmission) methods suggest that fracture apertures follow normal [e.g., Lee et al., 2003], lognormal [e.g., Keller, 1998; Keller et al., 1999], and gamma distributions [e.g., Tsang and Tsang, 1987], and can also be self-affine when facing surfaces are uncorrelated [Plouraboue et al., 1995]. The characterization of natural fracture wall surface morphology has shown that fracture wall roughness is well described by a self-affine scale invariance [Schmittbuhl et al., 1995; Bouchaud, 1997; Brown and Scholz, 1985]. It is noteworthy that the statistical characteristics of the aperture field are not necessarily the same as those of the wall surface [Roberds et al., 1990]. [3] Numerical studies [e.g., Tsang and Tsang, 1990], theoretical analyses [e.g., Gelhar, 1993], and experimental investigations [e.g., Keller et al., 1999] have shown that the mean (m b ), standard deviation (s b ) and correlation length (l) of a fracture aperture field can be used to predict both the transmissivity and the dispersion coefficient of fracture apertures. In addition to these factors, the anisotropy of both the aperture field and the fracture wall influence flow and 1 Department of Civil Engineering, McMaster University, Hamilton, Ontario, Canada. Copyright 2009 by the American Geophysical Union /09/2007JB005161$09.00 transport in variable-aperture fractures. Natural fracture walls often show anisotropic roughness, particularly in plumose structures [e.g., Brown and Scholz, 1985] and fault zones [e.g., Power et al., 1987], which may also be manifested in the aperture field [Thompson and Brown, 1991]. Thompson and Brown [1991] investigated the effect of anisotropic wall roughness on flow and transport in single fractures using numerical techniques, and concluded that the directional characteristics of the fracture wall play a more important role in determining fracture transport properties than the degree of wall roughness. This observation does not hold for the aperture field itself, however, as Lee et al. [2003] demonstrated that the effect of aperture field anisotropy is insignificant compared with the variability and correlation length of the aperture field in their laboratoryscale experiments. It should be noted, however, that the conclusion presented by Lee et al. [2003] was based on experimental results from five fracture replicas with a limited range of coefficients of variation (COV, s b /m b ) and anisotropic ratios (AR, l b x/l b y). [4] Recent scaling analyses [Roux et al., 1998], and experimental and numerical investigations [Plouraboue et al., 1998; Detwiler et al., 2000] have revealed that there exist three distinct dispersion regimes for solute transport in variable-aperture fractures, specifically the molecular diffusion regime, the geometric dispersion regime and the Taylor dispersion regime, which dominate within different ranges of the Peclet number, Pe: Pe ¼ vm b =D * where v [Lt 1 ] is the mean solute velocity, m b [L] is the arithmetic mean aperture, and D* [L 2 t 1 ] is the molecular diffusion coefficient. The range of Pe, in turn, is controlled ð1þ 1of12
3 by the mean, variance, and correlation structure of the aperture field. Detwiler et al. [2000] numerically delineated the range of Pe in which different dispersion mechanisms dominate for four isotropic aperture fields with the same arithmetic mean aperture by changing the hydraulic gradient. The effects of mean aperture (m b ) and anisotropy on the dominance of each dispersion regime were, however, beyond the scope of their investigation. Relatively few studies have been devoted to investigating the effects of aperture field heterogeneity and anisotropy on solute transport properties in single fractures, and therefore the understanding of the effects of these properties on transport remains limited. The effects of these parameters play a key role in furthering our understanding of solute transport in saturated fractured environments, which is required in order to build reliable predictive models. Fractured rock environments are important due to their prevalence throughout North America, and the speed at which contaminants can migrate through these environments relative to unconsolidated porous media. Additionally, transport in saturated fractured rock environments has not been investigated to the same extent that it has been in unconsolidated porous media environments, and therefore the understanding of these environments is still lacking significantly. Accurate and reliable predictive models are required in order to develop sound management strategies for fractured rock aquifers; to assess the risk to human and environmental health posed by the migration of contaminants in fractured environments; and to develop remedial strategies for contaminated fractured rock environments. [5] Therefore, the goal of this research is (1) to investigate the influence of an aperture field s arithmetic mean, variance, and AR on the predominance of each dispersion regime and (2) to systematically investigate the interactive influence of the mean, variance, and AR of fracture aperture fields on dispersivity in variable-aperture fractures. 2. Methods [6] To reach this goal, a 3 3 factorial experiment (Tables 1a and 1b) was designed to investigate the influence of the arithmetic mean (m b ), standard deviation (s b ), and anisotropy ratio (AR) of the aperture field on dispersivity and the predominance of each dispersion regime. Computer-generated aperture fields were employed, and aperture-averaged velocity fields were obtained by solving the Reynolds equation numerically. Solute transport in a single fracture was simulated using the random walk particle tracking (RWPT) technique Fracture Aperture Field Generation [7] It was assumed that aperture fields follow a lognormal distribution, and the anisotropic covariance function, C ln(b), of the log aperture has the form [Chrysikopoulos and James, 2003] 2 C lnðbþ ðrþ ¼s 2 ln b exp4 r2 x l 2 x þ r2 y l 2 y! ð2þ where b [L] is the fracture aperture, s 2 lnb is the variance of the ln aperture, lnb, r =(r x,r y ) T is a two-dimensional vector whose magnitude is the separation distance between two aperture measurements, and l x [L] and l y [L] are the correlation length scales of lnb in the x and y directions, respectively. The m b and s b are related to their log transformations by 1 s ln b ¼ ln s2 2 b m 2 þ 1 ð3þ b m ln b ¼ ln m b 0:5s 2 ln b The quasi-three-dimensional fracture aperture fields employed in this research are 16.0 m (x direction) by 2.0 m (y direction). Each aperture field was discretized into a grid of 400 elements (x direction) by 50 elements (y direction) such that each 4 cm by 4 cm element had a distinct aperture. The aperture fields were generated using SPRT2D [Gutjahr, 1989], which is based on the Fast Fourier Transform technique. Figure 1a shows a typical aperture field generated in this manner. Fifty realizations for each set of aperture field statistics (m b, s b, and AR) were generated by changing the seed number supplied to the random field generator. The random noise present in the stochastic simulations was smoothed out using ensemble averages of the breakthrough curves generated by each of the 50 aperture field realizations [James et al., 2005] Flow Field Calculation [8] The steady state flow field in each fracture plane was obtained by solving the Reynolds equation using a fully implicit finite difference technique [Chrysikopoulos and b3 ðx; þ b3 ðx; yþ y¼0:0 ¼ yþ y¼2:0 ¼ 0 hðx; yþj x¼0:0 ¼ h 0 hðx; yþj x¼16:0 ¼ ¼ 0 ð4þ ð5aþ ð5bþ ð5cþ ð5dþ ð5eþ where x [L] is the coordinate along the fracture length, y [L] is the coordinate along the fracture width, h(x,y) [L] is the local hydraulic head, and b(x,y) [L] is the local fracture aperture. The boundary conditions described in equations (5a) (5e) represent no-flow boundaries on the sides of a horizontal fracture, and constant head upstream and downstream boundaries. The equivalent aperture between adjacent elements is approximated by the harmonic mean 2of12
4 Table 1a. Factorial Experimental Design Run Mean m b (m) Standard Deviation s b (m) Coefficient of Variation (COV = s b /m b ) Longitudinal Correlation Length l x b (m) Transverse Correlation Length l y b (m) Anisotropic Ratio (AR = l x b /l y b ) aperture of the two elements [Reimus, 1995]. The mean fluid velocity components in both the x and y directions are calculated by [Chrysikopoulos and James, 2003] v x ¼ gb2 ðx; yþ 12h v y ¼ gb2 ðx; yþ yþ ð6aþ ð6bþ where g [ML 2 t 2 ] is the fluid specific weight, and h [ML 1 t 1 ] is the fluid dynamic viscosity. Figure 1b gives a schematic illustration of the coordinate system and the boundary conditions employed in these simulations. The model parameters are listed in Table 2. Further details regarding the finite difference modeling scheme for the flow field are included in the work by Chrysikopoulos and James [2003]. [9] It is well recognized that the Reynolds equation employs numerous simplifying assumptions, the most limiting of which is that inertial forces are negligible. Several studies have demonstrated that the Reynolds equation over predicts the flow rate by up to twice the actual value [e.g., Hakami and Larsson, 1996; Yeo et al., 1998]. However, Brush and Thomson [2003] conducted simulations to compare the Reynolds equation to both the Stokes and Navier Table 1b. Experimental Design for Isotropic Aperture Fields Run Mean m b (m) Standard Deviation s b (m) Coefficient of Variation (COV = s b /m b ) Longitudinal Correlation Length l x b (m) Transverse Correlation Length l y b (m) Anisotropic Ratio (AR = l x b /l b y) of12
5 Figure 1. (a) A contour plot of a fracture aperture field and (b) a schematic diagram of the coordinate system and boundary conditions for the flow and solute transport simulations. Stokes equations under various geometric and kinematic conditions, and concluded that the Reynolds equation may be considered valid when the following criteria are satisfied: Re < 1; Rehbi=l b < 1; Res b =hbi < 1 The further beyond these ranges the parameters get, the more the Reynolds equation will over predict the flow. These conditions are met in a majority of the simulations conducted here (except when COV = 1.2), and are never exceeded significantly. Therefore, it is sound to assume that the Reynolds equation is a reasonable approach for modeling the flow field Conservative Solute Transport [10] There are two theoretical approaches for modeling solute transport in single fractures: the Eulerian approach and the Lagrangian approach. In an Eulerian framework, partial differential equations governing solute transport are solved, generally using numerical techniques, to obtain the solute concentration distribution in space. However, these numerical methods are often plagued by numerical dispersion and artificial oscillation when applied to advectively dominated transport problems [Zheng and Bennett, 2002]. The RWPT method, which falls within the Lagrangian framework, offers an alternative to the Eulerian approach particularly when the Pe and dispersive anisotropy are large [Kinzelbach and Uffink, 1991]. Since the purpose of this research is to investigate the effects of aperture field variability and AR on dispersion, and any numerical dispersion could seriously influence the conclusions drawn from this work, the RWPT technique was chosen to simulate solute transport. [11] The RWPT technique has been applied in numerous studies to investigate contaminant transport in fractured media [e.g., Moreno et al., 1988; Detwiler et al., 2000; Chrysikopoulos and James, 2003; James et al., 2005]. In vector form, the RWPT equation is given by [Tompson and Gelhar, 1990] X n ¼ X n 1 þ AðX n 1 ÞDt þ BX n 1 pffiffiffiffiffi Z Dt where n refers to the time step, X n [L] is the threedimensional particle position vector at time ndt, A(X n 1 )is Table 2. Flow and Transport Simulation Parameters Parameter Value Flow Simulation Fracture dimensions 16.0 m 2.0 m (length width) Fracture discretization (elements elements) Element size (dx dy) 0.04 m 0.04 m Temperature 20 C Dynamic viscosity h Nms 2 Specific weight g 9789 N m 3 Hydraulic gradient 0.25/16 for m b = 100 mm 0.025/16 for m b = 300 mm, m b = 600 mm, and m b = 900 mm Molecular dispersion coefficient D* Time step Dt Transport Simulation m 2 s 1 1s ð7þ 4of12
6 the absolute forcing vector evaluated at X n 1, B(X n 1 )isa deterministic scaling second-order tensor evaluated at X n 1, and Z is a vector of three independent normally distributed random numbers with a mean of zero and a unit variance. [12] The same discretization scheme was employed for the solute transport simulations as for flow field simulations, and the aperture within each discretized element remained constant. It was assumed that the aperture field is symmetric about the center plane of the fracture. Upon calculation of the two-dimensional flow field, a parabolic velocity profile was imposed across the fracture and the quasi-three-dimensional RWPT equations were employed to simulate the transport process [James et al., 2005]: x n ¼ x n 1 þ v x x n 1 ; y n 1 ; z n 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Zð0; 1Þ 2D * Dt y n ¼ y n 1 þ v y x n 1 ; y n 1 ; z n 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Zð0; 1Þ 2D * Dt ( ) z 2 Dt bðx; yþ ( ) z 2 Dt bðx; yþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi z n ¼ z n 1 þ Zð0; 1Þ 2D * Dt ð8aþ ð8bþ ð8cþ where v x [Lt 1 ] and v y [Lt 1 ] are the local mean fluid velocities in the x and y directions, respectively, and D* [L 2 t 1 ] is the molecular diffusion coefficient. [13] The initial condition involves the instantaneous release of 10,000 conservative particles across the inlet of the fracture. The particles are then distributed on the basis of the local volumetric flow rate and velocity vectors as detailed by James et al. [2005]. The RWPT model parameters are listed in Table Dispersion in Variable Aperture Fractures [14] The hydrodynamic dispersion of a conservative tracer migrating through single fractures is governed by a combination of molecular diffusion (D*), Taylor dispersion (D Taylor ) [Taylor, 1953], and geometric dispersion (D G ) [Roux et al., 1998; Plouraboue et al., 1998; Detwiler et al., 2000]. Each of these dispersion regimes becomes dominant in different ranges of the Pe. Molecular diffusion dominates for Pe 1. Taylor dispersion, which results from the combined effects of molecular diffusion and the velocity distribution across the fracture aperture, is as follows for parallel plate fractures: D Taylor ¼ D * þ v2 b 2 210D * where b [L] is the local aperture, v [Lt 1 ] is the local mean velocity in the fracture and D* [L 2 t 1 ] is the molecular diffusion coefficient. In this work D Taylor was approximated for variable aperture fractures by replacing the local aperture ð9þ b [L] in equation (9) with the arithmetic mean aperture m b [L] [e.g., Detwiler et al., 2000]: D Taylor ¼ D * þ v2 m 2 b 210D * ¼ D* þ D* Pe 2 ð10þ Geometric dispersion results from the velocity variations induced by aperture field variability. The geometric dispersion coefficient, D G [L 2 t 1 ], is based on a stochastic analysis of solute transport in variable-aperture fractures and is given by Gelhar [1993]: D G ¼½3 þ gðs 2 ln b ÞŠs2 ln b lv ð11aþ g s 2 ln b ¼ 1 þ 0:205s 2 ln b þ 0:16s 4 ln b þ 0:045s6 ln b þ 0:0115s 8 ln b for 0 < s 2 ln b < 5 ð11bþ Gelhar s [1993] analysis was based on the assumptions that the logarithm of the fracture aperture (lnb) is a statistically stationary, isotropic, two-dimensional Gaussian random field, and that flow within a variable-aperture fracture can be modeled by the Reynolds equation. For the anisotropic aperture fields employed in this research, the correlation length, l, was replaced by the omnidirectional correlation length, l s, obtained by fitting a combination of the nugget effect and the spherical model to the semivariogram of the aperture field: 8 >< nugget þ A" # h > l s gðhþ ¼ nugget þ A 1:5H 0:5H 3 >: l s l 3 s h l s ð12þ where H [L] is the lag distance, A [L 2 ] is the scale of the spherical variogram model, and l s [L] is the range of the spherical model and represents the omnidirectional correlation length. [15] A fourth dispersion coefficient, the fitted dispersion coefficient (D f [L 2 t 1 ]), is employed in this work and represents the effective dispersion coefficient with contributions from D*, D Taylor and D G. D f was calculated by fitting the one-dimensional analytical solution for solute transport in parallel plate fractures to the numerically generated breakthrough curves. The one-dimensional mathematical model employed to simulate solute transport in parallel plate fractures is ¼ D 2 v Cð0; tþ ¼ M Q dðtþ ð13aþ ð13bþ Cðx; 0Þ ¼0 x > 0 ð13cþ lim Cðx; tþ ¼0 ð13dþ x!1 where C [ML 3 ] is the solute concentration, v [Lt 1 ]isthe mean solute transport velocity, M [M] is the mass of solute injected, Q [L 3 t 1 ] is the volumetric flow rate, x [L] is the 5of12
7 Figure 2. The numerically generated breakthrough curve from run 11 (m b = s b = AR = 5). The error bars represent one standard deviation of the 50 effluent breakthrough curves generated using these same aperture field statistics. spatial coordinate, t [t] is the time, and d(t) is the Dirac delta function for the time variable. [16] The analytical solution for equations (13a) (13d) is given by [Kreft and Zuber, 1978]! Cðx; tþ ¼ M x ðx vtþ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp Q 4pD f t 3 4D f t D f ¼ a L v þ D * ð14þ ð15þ where a L [L] is the longitudinal dispersivity. PEST (Watermark Numerical Computing, version 9.0) was employed for the fitting process, in which both v and a L were used as fitting parameters. 3. Results and Analysis [17] Tsang and Tsang [1990] calculated the dispersion for several realizations of aperture fields with a specific set of COV and l. They repeated this exercise for several sets of COV and l values in isotropic aperture fields, and found that the range for their measure of dispersion increased with increasing COV. The numerically generated breakthrough curves reported here, however, represent an average of 50 statistical realizations for each set of aperture field statistics (i.e., COV, l). Therefore, these breakthrough curves describe the central tendency of dispersion, and are not skewed on the basis of the COV. Figure 2 shows the numerically generated breakthrough curve from run 11. The error bars represent one standard deviation of the effluent concentration from the 50 aperture field realizations Influence of Aperture Field Statistics on the Dispersion Regime [18] Several researchers [Roux et al., 1998; Detwiler et al., 2000] have suggested that the overall hydrodynamic dispersion coefficient (D TL [L 2 t 1 ]) in variable-aperture fractures can be described by a sum of the Taylor and geometric dispersion coefficients as follows: D TL ¼ D Taylor þ D G ð16þ Therefore, the dominance of either the Taylor or geometric regime in the overall dispersion coefficient can be determined by a ratio of D Taylor to D G, based on equations (10), (11a), and (11b) as follows: D Taylor D G 1 ¼ 210 Pe a G a G ¼ s2 ln b l 3 þ g s2 ln b m b ð17aþ ð17bþ where a G is introduced to simplify equation (17a). Equation (17a) shows that D Taylor /D G is linearly dependent on Pe, with a slope of (1/210)/a G.IfD Taylor /D G > 1, the Taylor dispersion regime dominates, and if D Taylor /D G < 1, the geometric dispersion regime dominates. Together, equations (1), (17a), and (17b) clearly show that it is a combination of the hydraulic gradient and the aperture field characteristics that control the dispersion regime. [19] Figure 3 illustrates the dependence of D Taylor /D G on Pe, m b, and s b. It shows that for a constant value of (1/210)/ a G, D G becomes more dominant as Pe decreases, and when Pe is fixed, D G becomes more dominant as (1/210)/a G decreases. For a fixed mean aperture, D G becomes increasingly dominant as s b increases with the exception of the case where m b = 100 mm and s b = 120 mm. This particular case has the largest COV (1.2) of all those presented in this research. This inconsistency can be explained by the inadequacy of the Reynolds equation to simulate flow in 6of12
8 Figure 3. Dispersion regime dependence on Pe with respect to m b, s b and AR. Note that (1) the hydraulic head loss across different fractures is 0.25 m for m b = 100 mm and m for m b = 300 mm, m b = 600 mm, and m b = 900 mm. (2) The dashed lines correspond to various values of the slope, i.e., (1/210)/{s 2 lnb l[3 + g (s 2 lnb )]}/m b. It should be noted here that both D Talyor /D G and Pe are plotted on a log scale, and therefore, although these dashed lines appear parallel, they are not. fractures with COVs in this range. Brush and Thomson [2003] employed both the Reynolds and Stokes equations to predict flow in variable aperture fractures and found that the flow rate is increasingly overestimated by the Reynolds equation, with respect to the Stokes equation, as the COV increases. [20] Figure 3 also shows that for aperture fields following a lognormal distribution, the dominant dispersion regime is controlled by m b and, to a lesser degree, s b. An explanation to this phenomenon is provided by equation (18a), which was obtained by substituting the cubic law (equation (18b)) into equation (17a). D Taylor D G 1 ¼ 210 Pe a G ¼ g 12hD * ln s2 b m 2 þ 1 b m4 b l 3 þ g v ¼ g 12h m2 b ln s2 b m 2 þ 1 b Dh DL Dh DL ð18aþ ð18bþ Since it is well recognized that the cubic law does not describe fluid flow in variable-aperture fractures accurately, the information derived from equation (18a) must be considered qualitative. An examination of equation (18a) indicates, however, that m b has a larger influence on the value of D Taylor /D G than s b. [21] Figure 3 also illustrates the relationship between the dispersion regime, m b, and AR. It can be seen that if all else is held constant, D G has a slightly larger influence for anisotropic aperture fields than it does for isotropic aperture fields. However, Figure 3 clearly shows that the AR has a much smaller influence on the dispersion regime than s b. [22] Figure 4a shows the numerically generated breakthrough curve from run 11 fit to the 1-D analytical solution for solute transport (equation (14)). Figure 4b shows the magnitude of D f, determined from fitting equation (14) to the numerically generated breakthrough curves, compared with D TL, calculated by equation (16). For comparison purposes, eight additional simulations were conducted in isotropic aperture fields. It was found that D f is typically between 2% and 60% smaller than D TL, with the exception of a few simulations with m b = 100 mm, where D f is between 2 and 37% larger than D TL. Two possible explanations for this discrepancy are (1) errors associated with the first-order approximations made by the stochastic analysis in deriving equation (16) [Keller et al., 1999] and (2) errors introduced by solving the Reynolds equation to obtain the threedimensional velocity field, which required the imposition of a local parabolic velocity profile across the aperture. Flow simulations in variable-aperture fractures based on the Stokes and Navier-Stokes equations, have shown that velocity profiles, in fact, deviate from the ideal parabolic distribution [Brush and Thomson, 2003]. Equation (10) has, in fact, been employed to represent the Taylor dispersion coefficient in variable-aperture fractures by others [e.g., Detwiler et al., 2000]; however, further theoretical and numerical analyses are required in order to confirm the validity of this application. Figure 4 confirms the validity of using the omnidirectional correlation length in equation (16) to estimate the dispersion coefficient, as no significant 7of12
9 Figure 4. (a) The fit between the numerically generated breakthrough curve from run 11 (m b = m, s b = m, AR = 5) with the 1-D analytical solution for solute transport and (b) a comparison of D f with D TL. difference between the isotropic and anisotropic aperture fields were notable in the comparison between D f and D TL Influence of m b, s b, and AR on a L [23] In this section, the dispersivity refers to that obtained through the fitting technique. Equation (15) assumes that D f is linearly dependent on v and therefore a L is solely a function of aperture field statistics. Figure 5 shows dispersivity plotted against the COV for a range of ARs. Table 3 presents the ratio of the largest to the smallest dispersivities calculated for each COV. Several trends emerge upon the examination of Figure 5 and Table 3. First, in general, dispersivity increases with increasing COV, which is consistent with equations (11a) and (11b). However, the dispersivities for cases of m b = 900 mm are significantly larger than those with the same COV but different mean aperture. This phenomenon can be explained through a change in the dominant dispersion regime. As shown in Figure 3, D G dominates in fractures with m b of 100, 300, and 600 mm, and therefore the relationship between the dispersion coefficient and the velocity is linear (equation (11a)). Conversely, in fractures with a mean aperture of 900 mm, D Taylor dominates, and the dispersion coefficient is proportional to the square of the velocity (equation (10)). However, the fitting procedure employed to calculate the dispersivity values in Figure 5 assumes a linear relationship between the dispersion coefficient and the velocity (equation (15)). We postulate, therefore, that it is not appropriate to assume a linear dependence of dispersion coefficient on velocity when D Taylor dominates, as the result will be erroneously large. [24] Second, the difference between dispersivities due to the AR is statistically insignificant for aperture fields with small COVs and the three ARs considered in this work; however, the spread in dispersivity for these ARs increases as the COV increases. This implies that the effect of AR on dispersivity is significant only when the COV is large (Figure 5). Table 3 indicates that the effect of the AR on dispersivity also depends on the m b. For the same COV, the ratio of the anisotropic dispersivity (a L,AR = 5.0, a L,AR = 0.2 )to isotropic dispersivity (a L,AR = 1.0 ) is larger for fractures with smaller m b (Table 3). Lee et al. [2003] investigated the influence of COV, l, and AR on dispersivity through laboratory-scale experiments, and concluded that the effect of AR is insignificant compared with those of the COVand l. However, their conclusion was reached by comparing the dispersivities obtained from the moment method of breakthrough curve analysis [Yu et al., 1999] with those calculated from equations (11a) and (11b), in which l for isotropic aperture fields was replaced by an omnidirectional correlation length for anisotropic aperture fields. It is not entirely appropriate to evaluate the effect of AR on dispersivity through this comparison, however, as the fact that there were only small differences (at most 32%) between the dispersivities obtained from the moment method and equations (11a) and (11b) only indicates that an effective correlation length for l in equations (11a) and (11b) can be employed to approximate the dispersivity. Figure 6 shows the dispersivities obtained by Lee et al. [2003] through the fitting technique plotted against COV, and illustrates that the data from their experiments were insufficient to reach a definite conclusion with respect to the effect of AR on dispersivity. [25] The finding that anisotropy is significant only when the COV is large has important implications for numerical modeling. For real problems, it is difficult to obtain a dispersion tensor for single anisotropic fractures, and therefore isotropic dispersion tensors are typically employed in modeling practice. This finding suggests that such a simplification provides a reasonable estimation of dispersivity as long as the aperture field variability is not large. The data presented in Figures 5a and 5b suggest that a COV of approximately 0.2 may be an appropriate cutoff for simplifying anisotropic apertures with isotropic dispersion tensors; however, more work is required to confirm this suggestion. [26] It was further noted from this work that for aperture fields with different ARs, the relative magnitude of dispersivity changes with the COV as follows: a L;AR¼0:2 > a L;AR¼5:0 > a L;AR¼1:0 for 0:4 < COV 1:2 a L;AR¼5:0 > a L;AR¼0:2 > a L;AR¼1:0 for 0:08 COV 0:4 ð19aþ ð19bþ 8of12
10 Figure 5. Dispersivity versus COV for (a) all 27 fracture aperture fields investigated in this research and (b) enlarged view showing aperture fields with small COVs. Equations (19a) and (19b) show that for a single fracture with the same m b and s b, those with anisotropic aperture fields (either AR > 1 or AR < 1) have higher dispersivities than those with isotropic aperture fields. This is due to the fact that for aperture fields with the same COV, anisotropic aperture fields lead to a larger spatial variation in fluid velocity, which therefore results in larger dispersivities. However, it is not clear why the relative magnitude of a L,AR = 5.0 and a L,AR = 0.2 depends on the COV. 9of12
11 Table 3. Effect of AR and COV on Dispersivity a COV ða L;AR¼5:0 þa L;AR¼0:2 Þ=2 a L;AR¼ (for m b = 900 mm) (for m b = 600 mm) (for m b = 900 mm) (for m b = 900 mm), 1.64 (for m b = 600 mm), 1.83 (for m b = 300 mm) (for m b = 600 mm) (for m b = 300 mm) (for m b = 100 mm), 2.25 (for m b = 300 mm) (for m b = 100 mm) (for m b = 100 mm) a The a L,AR = 0.2, a L,AR = 5.0, and a L,AR = 1.0 represent the dispersivity corresponding to the ARs of 0.2, 5.0, and 1.0, respectively. [27] Thompson [1991] investigated the effect of COV on dispersivity for fractures with isotropic walls, and Thompson and Brown [1991] calculated the longitudinal dispersivity as a function of fracture wall anisotropy and aperture field COV. Their observations support those from this work that the dispersivity increases with the COV. The spread of dispersivity, however, due to the AR of the aperture field observed in this work is not consistent with the spread in dispersivity observed by Thompson and Brown [1991] due to the AR of fracture walls. This discrepancy is likely due to the fact that the AR of the fracture walls has a different effect on dispersivity than the AR of the aperture field itself. [28] Figure 7 shows D f plotted against COV for a range of ARs and reveals a different trend than that of Figure 5. For a fixed m b, the dispersion coefficient, as well as the spread in the dispersion coefficient with different ARs, increases with increasing values of the COV, with the exception of the cases where m b = 100 mm and s b = 120 mm. This is primarily due to the inadequacy of the Reynolds equation for larger COV, as discussed in section 3.1. However, for fractures with different m b, a larger COV does not necessarily correspond to a larger dispersion coefficient. This is due to the fact dispersion coefficient depends on both dispersivity and fluid velocity; and m b plays a more important role than s b in determining the fluid velocity. Although the dispersivities increase as the COV increases (equation (11a)), the decrease in m b causes a decrease in the velocity to a larger extent. The resulting dispersion coefficients (equation (15)) therefore decrease with decreasing m b. 4. Summary [29] A3 3 factorial experiment was designed to investigate the effect of aperture field statistics (i.e., m b, s b, and AR) on dispersivity and dispersion regimes numerically. Flow fields were calculated using Reynolds equation, and a two-dimensional random walk particle tracking technique was used to simulate solute transport. It was demonstrated that the arithmetic mean aperture m b and, to a lesser degree, the standard deviation s b, influence the value of D Taylor /D G., and hence the dominant dispersion regime. When the hydraulic gradient is fixed, D G accounts for a larger fraction of D TL for aperture fields with a larger COV. The influence of AR on dispersion regimes, however, seems insignificant for the correlation structure of the aperture fields employed in this research. Additional simulation studies using aperture fields with a larger range of correlation lengths and ARs would help to clarify the effect of anisotropy on dispersion regime. Figure 6. Dispersivities versus COV for different ARs (data from Lee et al. [2003]). 10 of 12
12 Figure 7. Dispersion coefficient versus COV for a range of ARs. Note that the hydraulic head loss across different fractures is 0.25 m for m b = 100 mm and m for m b = 300 mm, m b = 600 mm, and m b = 900 mm. [30] The results also revealed the interactive effects of the COV and the AR on dispersivity in fractures. It was found that the dispersivity, as well as the spread of dispersivity, increases as the COV increases. The effect of the AR is significant only when the COV is large. Therefore, if the COV is not large (]0.2), the isotropic dispersivity tensor can be used to replace the anisotropic dispersivity tensor in numerical models simulating solute transport in variableaperture fractures without introducing unacceptable errors. [31] Acknowledgments. This research was funded by the NSERC Discovery Grant Program (S. Dickson and Y. Guo). The authors would like to thank S. C James for generously providing the code employed in this research. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET, References Bouchaud, E. (1997), Scaling properties of crack surfaces, J. Phys. Condens. Matter, 9, , doi: / /9/21/002. Brown, S. R., and C. H. Scholz (1985), Broad bandwidth study of the topography of natural rock surfaces, J. Geophys. Res., 90, 12,575 12,582, doi: /jb090ib14p Brush, D. J., and N. R. Thomson (2003), Fluid flow in synthetic roughwalled fractures: Navier-Stokes, Stokes, and local cubic law simulations, Water Resour. Res., 39(4), 1085, doi: /2002wr Chrysikopoulos, C. V., and S. C. James (2003), Transport of neutrally buoyant and dense variably sized colloids in a two-dimensional fracture with anisotropic aperture, Transp. Porous Media, 51, , doi: /a: Detwiler, R. L., H. Rajaram, and R. J. Glass (2000), Solute transport in variable-aperture fractures: An investigation of the relative importance of Taylor dispersion and macrodispersion, Water Resour. Res., 36(7), , doi: /2000wr Gelhar,L.W.(1993),Stochastic Subsurface Hydrology, pp , Prentice-Hall, Old Tappan, N. J. Gentier, S., E. Lamontagne, G. Archambault, and J. Riss (1997), Anisotropy of flow in a fracture undergoing shear and its relationship to the direction of shearing and injection pressure, Int. J. Rock Mech. Min. Sci., 34(3), 412, doi: /s0148. Gutjahr, A. L. (1989), Fast Fourier transform for random field generation, Rep. 4-R R, N. M. Inst. of Min. and Technol., Socorro. Hakami, E., and E. Larsson (1996), Aperture measurement and flow experiments on a single natural fracture, Int. J. Mech. Min. Sci. Geomech. Abstr., 33(4), , doi: / (95) James, S. C., T. K. Bilezikjian, and C. V. Chrysikopoulos (2005), Contaminant transport in a fracture with spatially variable aperture in the presence of monodisperse and polydisperse colloids, Stochastic Environ. Res. Risk Assess., 19(4), , doi: /s Keller, A. (1998), High resolution, non-destructive measurement and characterization of fracture apertures, Int. J. Rock Mech. Min. Sci., 35(8), , doi: /s (98) Keller, A. A., P. V. Roberts, and M. J. Blunt (1999), Effect of fracture aperture variations on the dispersion of contaminants, Water Resour. Res., 35(1), 55 63, doi: /1998wr Kinzelbach, W., and G. Uffink (1991), The random walk method and extensions in groundwater modeling, in Transport Processes in Porous Media, edited by J. Bear and M. Y. Corapcioglu, pp , Kluwer Acad., Boston, Mass. Kreft, A., and A. Zuber (1978), On the physical meaning of dispersion equation and its solutions for different initial and boundary conditions, Chem. Eng. Sci., 33, , doi: / (78) Lee, J., J. M. Kang, and J. Choe (2003), Experimental analysis on the effects of variable apertures on tracer transport, Water Resour. Res., 39(1), 1015, doi: /2001wr Meheust, Y., and J. Schmittbuhl (2001), Geometrical heterogeneities and permeability anisotropy of rough fractures, J. Geophys. Res., 106, , doi: /2000jb Moreno, L., Y. W. Tsang, C. F. Tsang, F. Hale, and I. Neretnieks (1988), Flow and tracer transport in a single fracture: A stochastic model and its relation to some field observations, Water Resour. Res., 24(12), , doi: /wr024i012p Plouraboue, F., P. Kurowski, J. P. Hulin, S. Roux, and J. Schmittbuhl (1995), Aperture of rough crack, Phys. Rev. E, 51, , doi: /physreve Plouraboue, F., J. P. Hulin, S. Roux, and J. Koplik (1998), Numerical study of geometric dispersion in self-affine rough fractures, Phys. Rev. E, 58, , doi: /physreve Power, W. L., T. E. Tullis, S. R. Brown, G. N. Boitnott, and C. H. Scholz (1987), Roughness of natural fault surfaces, Geophys. Res. Lett., 14, 29 32, doi: /gl014i001p Reimus, P. W. (1995), The use of synthetic colloids in tracer transport experiments in saturated rock fractures, Rep. LA T, Los Alamos Natl. Lab., Los Alamos, N. M. Roberds, W. J., M. Iwano, and H. H. Einstein (1990), Probabilistic mapping of rock joints surface, in Rock Joints, edited by N. Barton and O. Stephansson, pp , A. A. Balkema, Brookfield, Vt. 11 of 12
13 Roux, S., F. Plouraboue, and J. Hulin (1998), Tracer dispersion in rough open cracks, Transp. Porous Media, 32, , doi: / A: Schmittbuhl, J., F. Schmitt, and C. H. Scholz (1995), Scaling invariance of crack surfaces, J. Geophys. Res., 100, , doi: / 94JB Taylor, G. (1953), Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. London, Ser. A, 219, Thompson, M. E. (1991), Numerical simulation of solute transport in rough fractures, J. Geophys. Res., 96, , doi: /90jb Thompson, M. E., and S. R. Brown (1991), The effect of anisotropic surface roughness on flow and transport in fractures, J. Geophys. Res., 96, 21,923 21,932, doi: /91jb Tompson, A. F. B., and L. W. Gelhar (1990), Numerical simulation of solute transport in three-dimensional randomly heterogeneous porous media, Water Resour. Res., 26(10), Tsang, Y. W., and C. F. Tsang (1987), Channel model of flow through fractured media, Water Resour. Res., 23(3), , doi: / WR023i003p Tsang, Y. W., and C. F. Tsang (1990), Hydrological characterization of variable-aperture fractures, in Rock Joints, edited by N. Barton and O. Stephansson, pp , A. A. Balkema Publishers, Brookfield, V. T., USA. Yeo, I. W., M. H. DeFreitas, and R. W. Zimmerman (1998), Effect of shear displacement on the aperture and permeability of a rock fracture, Int. J. Rock Mech. Min. Sci., 35(8), , doi: /s (98)00165-x. Yu, C., A. W. Warrick, and M. H. Conklin (1999), A moment method for analyzing breakthrough curves of step inputs, Water Resour. Res., 35(11), , doi: /1999wr Zheng, C., and G. D. Bennett (2002), Applied Contaminant Transport Modeling, John Wiley and Sons, Inc., New York. S. Dickson, Y. Guo, and Q. Zheng, Department of Civil Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L7, Canada. (sdickso@mcmaster.ca) 12 of 12
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